We show that \((ii)\)  implies \((i)\) .
Let \((X,\mu ,T)\)  be an ergodic \(k\) -step nilsystem for some \(k\in {\mathbb {N}}\) .
By [1]} we have that if \(f\in L^\infty (\mu )\)  is such that \(\vert \!|\!| f|\!|\!\vert _{k+1}=0\) ,
then \(f=0\) . Hence, any collection of sequences is good for seminorm estimates for \((X,\mu ,T)\) . Since, by assumption the
sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution for \((X,\mu ,T)\) , it follows from Theorem REF  that they are jointly ergodic for \((X,\mu ,T)\) .
